2005/09/08 by J. Dodziuk, Dodziuk, J.
Computer Science · Mathematics · #35K15 #39A12 #39A70 #58J50 #Advanced Mathematical Modeling in Engineering #Differential Geometry (math.DG) #FOS: Mathematics #Graph theory and applications #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #math.DG #math.SP #msc:35K15 #msc:39A12 #msc:39A70 #msc:58J50
paper · pdf · doi:10.48550/arxiv.math/0509193
14 pages, submitted to Proceedings of the conference "Krzysztof Wojciechowski 50 years - Analysis and Geometry of Boundary Value Problems," Roskilde, Denmark
arxiv created 2005/09/08 · openalex publication_date 2005/09/08 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We present some applications of ideas from partial differential equations and differential geometry to the study of difference equations on infinite graphs. All operators that we consider are examples of "elliptic operators" as defined by Y. Colin de Verdiere. For such operators, we discuss analogs of inequalities of Cheeger and Harnack and of the maximum principle (in both elliptic and parabolic versions), and apply them to study spectral theory, the ground state and the heat semigroup associated to these operators.